Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,35)(28,39)(29,33)(30,43)(31,37)(32,41)(34,42)(36,49)(38,45)(40,54)(44,56)(46,55)(47,51)(48,52)(50,53);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,44)(33,35)(34,45)(39,53)(40,50)(41,51)(42,52)(43,48)(46,47)(49,54)(55,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,41,29,40,42,55)(26,32,34,39,33,46,43,54)(27,36,38,51,37,50,52,56)(31,44,45,53,35,47,48,49);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(26,33)(27,37)(28,40)(34,43)(38,52)(41,55)(44,47)(49,53);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,40)(32,46)(36,50)(39,54)(41,55)(44,47)(49,53)(51,56)Polycyclic presentation:
2^9|b^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820462);